Vertices In Total Research Articles

Compact and Efficient Topological Mapping for Large-Scale Environment with Pruned Voronoi Diagram

Topological maps generated in complex and irregular unknown environments are meaningful for autonomous robots’ navigation. To obtain the skeleton of the environment without obstacle polygon extraction and clustering, we propose a method to obtain high-quality topological maps using only pure Voronoi diagrams in three steps. Supported by Voronoi vertex’s property of the largest empty circle, the method updates the global topological map incrementally in both dynamic and static environments online. The incremental method can be adapted to any fundamental Voronoi diagram generator. We maintain the entire space by two graphs, the pruned Voronoi graph for incremental updates and the reduced approximated generalized Voronoi graph for routing planning requests. We present an extensive benchmark and real-world experiment, and our method completes the environment representation in both indoor and outdoor areas. The proposed method generates a compact topological map in both small- and large-scale scenarios, which is defined as the total length and vertices of topological maps. Additionally, our method has been shortened by several orders of magnitude in terms of the total length and consumes less than 30% of the average time cost compared to state-of-the-art methods.

TOTAL EDGE AND VERTEX IRREGULAR STRENGTH OF TWITTER NETWORK

Twitter data can be converted into a graph where users can represent the vertices. Then the edges can be represented as relationships between users. This research focused on determining the total edge irregularity strength (tes) and the total vertices irregularity strength (tvs) of the Twitter network. The value could be determined by finding the greatest lower bound and the smallest upper bound. The lower bound was determined by using the properties, characteristics of the Twitter network graph along with the supporting theorems from previous studies, while the upper bound is determined through the construction of the total irregular labeling function on the Twitter network. The results in this study are the tes(TW)=18 and tvs(TW)=16.

Quantum search of many vertices on the joined complete graph

The quantum search on the graph is a very important topic. In this work, we develop a theoretic method on searching of single vertex on the graph [Phys. Rev. Lett. 114 110503 (2015)], and systematically study the search of many vertices on one low-connectivity graph, the joined complete graph. Our results reveal that, with the optimal jumping rate obtained from the theoretical method, we can find such target vertices at the time , where N is the number of total vertices. Therefore, the search of many vertices on the joined complete graph possessing quantum advantage has been achieved.

THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2

Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if alpha(G)=2 then G is a disconnected graph, otherwise G is a connected graph, G contains k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph.

On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

Shortest path planning for a tethered robot

We consider the problem of finding the shortest path for a tethered robot in a planar environment with polygonal obstacles of n total vertices. The robot is attached to an anchor point by a tether of finite length. The robot can cross the tether; i.e., the tether can be self-intersecting. Neither the robot nor the tether may enter the interior of any obstacle. The initial tether configuration is given as a polyline of k vertices.If the tether is automatically retracted and kept taut, we present an O(kn2log⁡n) time algorithm to find the shortest path between the source and the destination point. This improves the previous O(lkn3) time algorithm [24], where l is the number of loops in the initial tether configuration. If the tether can only be retracted while the robot backtracks along the tether, we present an algorithm to find the shortest path in O((n+log⁡k)log⁡n) time.

Brownian motion on ℝ-trees

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as locally infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact R \mathbb {R} -tree ( T , r ) (T,r) equipped with a Radon measure ν \nu on ( T , B ( T ) ) (T,{\mathcal B}(T)) . We specify a criterion under which the Brownian motion is recurrent or transient. For compact recurrent R \mathbb {R} -trees we provide bounds on the mixing time.

A new algorithm of inverse lithography technology for mask complexity reduction

A new complexity penalty term called the global wavelet penalty is introduced, which evaluates the high-frequency components of masks more profoundly by applying four distinctive Haar wavelet transforms and choosing the optimal direction on which the highest frequency components of the mask will be removed. Then, a new gradient-based inverse lithography technology (ILT) algorithm is proposed, with the computation of the global wavelet penalty as the emphasis of its first phase for mask complexity reduction. Experiments with three typical 65 nm flash ROM patterns under existing 90 nm lithographic conditions show that compared with the gradient-based algorithm, which relies on the so-called local wavelet penalty, the total vertices of the three results created by the proposed algorithm can be reduced by 12.89%, 12.63% and 12.64%, respectively, while the accuracy of the lithography results remains the same.

On total domination and support vertices of a tree

The total domination number γt(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of Chellali and Haynes [1]. In addition, we prove some bounds on the total domination ratio of trees.

Influence of the physical parameters of fields and of crop yield on the effective field capacity of a self-propelled forage harvester

Data for the effective field capacity of two self-propelled forage harvesters were collected from 163 fields located in Galicia, Northwest Spain, during the silage maize-harvesting season by using a telemetry system. The values obtained from the study were related to crop yield, field area, field longitudinal slope and cross slope, and to attributes of field shape: number of total vertices, number of acute vertices and shape index. Because of the lack of a shape index that reliably reflects the impact of field shape on the effective field capacity of harvesters, two new shape indices are proposed in this study. The results obtained using these two indices are analysed and compared with the results obtained using the conventional shape index (area/perimeter squared). The results of the analysis suggest a strong correlation between effective field capacity and both crop yield and field area. In addition, significant but weak correlations were found between effective field capacity and both longitudinal slope and shape index. The results obtained by considering all fields and just the largest fields (of more than 1 ha) were similar. Fields smaller than 1 ha showed a worse fit of the regression line to the data, which suggests the need to include new variables in the model. The results of the shape indices developed in this study were slightly better than the results obtained using the conventional index.

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to metrize the space of rooted compact real trees. Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields.

On fat partitioning, fat covering and the union size of polygons

The complexity of the contour of the union of simple polygons with n vertices in total can be O( n 2) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O( n log log n), which is a generalization of a similar result for fat triangles (Matoušek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O( n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.

Trellis decoding complexity of linear block codes

In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of uniform optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths.

A compact piecewise-linear voronoi diagram for convex sites in the plane

In the plane the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites arek disjoint convex sets we give a compact representation of the Voronoi diagram, usingO (k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons withn total vertices given in standard representations, we compute this diagram optimally in ź (k logn) deterministic time for the Euclidean metric and inO (k logn logm) deterministic time for the convex distance function defined by a convexm-gon.

Assembly sequences for polyhedra

The problem of finding sequences of motions for the assembly of a given object consisting of polyhedral parts arises in assembly planning. We describe an algorithm to compute the set of all translations separating two polyhedra withn vertices inO(n4) steps and show that this is optimal. Given an assembly ofk polyhedra with a total ofn vertices, an extension of this algorithm identifies a valid translation and removable subassembly inO(k2n4) steps if one exists. Based on the second algorithm, a polynomial-time method for finding a complete assembly sequence consisting of single translations is derived. An implementation incorporates several changes to achieve better average-case performance; experimental results obtained for simple assemblies are described.

Counting triangle crossings and halving planes

Every collection oft≥2n 2 triangles with a total ofn vertices in ℝ3 has Ω(t 4/n 6) crossing pairs. This implies that one of their edges meets Ω(t 3/n 6) of the triangles. From this it follows thatn points in ℝ3 have onlyO(n 8/3) halving planes.

Polyhedral line transversals in space

Algorithms are developed for determining if a set of polyhedral objects inR 3 can be intersected by a common transversal (stabbing) line. It can be determined inO(n logn) time if a set ofn line segments in space has a line transversal, and such a transversal can be found in the same time bound. For a set of polyhedra with a total ofn vertices, we give anO(n 4 logn) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.

Lower bounds on moving a ladder in two and three dimensions

It is shown that Ω(n 2) distinct moves may be necessary to move a line segment (a “ladder”) in the plane from an initial to a final position in the presence of polygonal obstacles of a total ofn vertices, and that Ω(n 4) moves may be necessary for the same problem in three dimensions. These two results establish lower bounds on algorithms that solve the motion-planning problems by listing the moves of the ladder. The best upper bounds known areO(n 2 logn) in two dimensions, andO(n 5 logn) in three dimensions.